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Overview

In the following notebok we visualize elliptic LCS in the Agulhas region from the AVISO dataset using the Polar Rotation Angle (PRA). The notebook is structured as follows:

  1. Import data from the file 'AVISO.mat' stored in the folder 'data'.
  2. Define computational parameters (such as the number of cores) and data.
  3. Define spatio-temporal domain.
  4. Interpolate velocity from the (discrete) gridded data.

  5. Polar Rotation Angle (PRA):

    • Compute gradient of flow map ∇Ft0tN(x0) over meshgrid using an auxiliary meshgrid.

    • Compute left and right singular vectors ξ1,2(x0),η1,2(x0) of the gradient of the flow map ∇Ft0tN(x0).

    • Compute PRA as:

      (1)PRAt0tN(x0)=cos−1⁡(⟨η2(x0),ξ2(x0)⟩)=cos−1⁡(⟨η1(x0),ξ1(x0)⟩).

      As the eigenvector associated to the maximum singular value of ∇Ft0tN(x0) is numerically more stable we compute the PRA using ηi,ξi:

      (2)PRAt0tN(x0)=cos−1⁡(⟨η2(x0),ξ2(x0)⟩).

Import Data

Computational parameters and data

Here we define the computational parameters and the data.

Spatio-temporal domain

Here we define the spatio-temporal domain over which to consider the dynamical system.

Velocity Interpolation

In order to evaluate the velocity field at arbitrary locations and times, we interpolate the discrete velocity data. The interpolation with respect to time is always linear. The interpolation with respect to space can be chosen to be "cubic" or "linear".

Polar Rotation Angle (PRA)

Next, we compute the PRA over the meshgrid over the given time-interval. We iterate over all initial conditions and first calculate the gradient of the flow map using an auxiliary grid. 'aux_grid' specifies the ratio between the auxiliary grid and the original meshgrid. This parameter is generally chosen to be between [15,110].

Elliptic LCS are identified as elliptic islands around local extrema in the PRA field. The elliptic LCSs are clearly visible as concentric closed contours of the PRA at time t0. These elliptic islands clearly distinguish vortical regions from the remaining flow. Note that the PRA highlights the same vortical flow structures as other commonly used rotation diagnostics such as the Trajectory Rotation Average, the EllipticLCS or the LAVD.

References

[1] Farazmand, M., & Haller, G. (2016). Polar rotation angle identifies elliptic islands in unsteady dynamical systems. Physica D: Nonlinear Phenomena, 315, 1-12.